Automatic derivative evaluation in solving boundary value problems: the renal medulla

Wexler, A.; Kalaba, Robert E.; Marsh, Donald J.

doi:10.1152/ajprenal.1986.251.2.f358

Cited by 7 publications

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“…Consequently, the model (2)- (34) is equivalent to a model with a Single descending limb. The index j in Equations (2)- (6), (26), in the transmural DLH fluxes of (24), (25), and in the DLH flows of (27) can be dropped. Thus the number of differential equations can be reduced to 27.…”

Section: Cf5(s)mentioning

confidence: 99%

“…The Systems modeling technique used in this section has been employed previously by several authors to stud[y various aspects of the integrated renal fimction [1,4,7,8,[12][13][14][15][16][17][18][19][20]. Finite-difference methods [12-14, 16, 20-23], quasilinearization [1,[24][25][26][27], invariant imbedding [27][28][29][30][31], and multiple shooting [4,17,32] have been suggested for the numerical Solution of the resulting boundary-value problems. A semidiscrete method combining collocation and shooting has been given in [19,33].…”

Section: Cf5(s)mentioning

confidence: 99%

“…The computation of the functions fc^ is completed by integrating the differential equations (3), (8), (17), (25) from s = 10.5 to s -oj. This together with the boundary conditions (27), (33) Note that a kN+l j = C A 6/ (10.5;a,p) and a^m + i,/ = C A 6 /(4.5;a,p).…”

Section: Breinbauer and P Lorymentioning

confidence: 99%

“…. (2)- (22), (24), (25) subject to the boundary conditions (26)- (31), (33), (34), where the concentrations C el in the right-hand sides of the differential equations and in (34) are replaced by the trial funetions C A6l (l -1,2). Note that aecording to Remark 2.1 the index j in the DLH flows can be dropped.…”

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The kidney model as an inverse problem

Breinbauer¹,

Lory

1991

Applied Mathematics and Computation

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The mammalian kidney is modeled by a multipoint boundary-value problem for a System of nonlinear ordinary differential equations. A corresponding inverse problem is presented, which allows the rigorous judgement of the potential of the given modeling technique. For its numerical Solution a discretization is proposed, which is tailor-made for kidney models. It leads to a nonlinear-programming problem with nonlinear equality and inequality constraints. The suggested methods are applied to current research problems in renal physiology.

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Section: Cf5(s)mentioning

confidence: 99%

Section: Cf5(s)mentioning

confidence: 99%

Section: Breinbauer and P Lorymentioning

confidence: 99%

mentioning

confidence: 99%

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The kidney model as an inverse problem

Breinbauer¹,

Lory

1991

Applied Mathematics and Computation

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A dynamic numerical method for models of renal tubules

Layton

Pitman

1994

Bltn Mathcal Biology

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We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.

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Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism

Thomas

1991

Bltn Mathcal Biology

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Modelling studies have played an important role in research on the mechanism of urine concentration and dilution by the medulla of the kidney ever since Hargitay and Kuhn (1951, Z. Elektrochem. 55, 539-558) first proposed that the parallel tubular structures in the kidney medulla must function as a "countercurrent multiplication" system. Present-day models, in keeping with our considerably improved understanding of most aspects of medullary structure-function relationships, have evolved into rather sophisticated systems of parallel tubes. In spite of this increasing complexity, it has remained the case that "model medullas" do not concentrate as well as the real kidney, especially in the inner medulla where only passive, diffusional transport occurs. Inasmuch as these models take into account the majority of contemporary ideas making up our global hypothesis about the functioning of this system, their failure to behave physiologically indicates that our understanding remains incomplete. The purpose of the present modelling study was to evaluate the implications of some recent measurements showing that permeabilities of NaCl (Ps) and urea (Pu) vary along the length of the descending thin limbs of Henle (Imai et al., 1988, Am. J. Physiol. 254, F323-F328), rather than being constant throughout this segment as had been assumed earlier. It was hoped that these newly measured values might explain, by a passive, diffusional process, the net solute addition at the bend of Henle's loop observed under some circumstances and heretofore attributed (though without any supporting experimental evidence) to active transport into the descending limb. The results of the present study show that whereas incorporation of the new values for Ps and Pu in the descending limbs of short nephrons does indeed improve the concentrating power of the model, these new values are nonetheless not sufficient to allow the model to build an osmolarity gradient that increases all the way through the inner medulla. This failing, which is common to virtually all modelling studies to date using measured values from rat kidneys, probably points to a key role for preferential exchange supposed by some to exist among certain tubule segments within vascular bundles in species whose kidneys have the highest concentrating power.

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Automatic derivative evaluation in solving boundary value problems: the renal medulla

Cited by 7 publications

References 0 publications

The kidney model as an inverse problem

The kidney model as an inverse problem

A dynamic numerical method for models of renal tubules

Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism

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